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For a field $F$ of characteristic $p$, the exact sequence $0\to {\mathbb{F}}_p\to F^{\mathrm{sep}}\stackrel{z\mapsto z^p-z}{\to }{F}^{\mathrm{sep}}\to 0$, together with additive normal-basis acyclicity, gives $H^2(\mathrm{Gal}(F^{\mathrm{sep}}/F),{\mathbb{F}}_p)=0$. Consequently every central finite embedding problem with group homomorphism kernel $C_p$ has a continuous weak solution: its pulled-back two-cocycle is a coboundary, whose correcting one-cochain supplies the lift. Over ${\mathbb{F}}_p((t))$, independent Artin–Schreier classes $t^{-m}$, with $m>0$ prime to $p$, allow the lift to be twisted by characters so that inertia maps onto the group homomorphism kernel as well as onto the original quotient. Induction through central order-$p$ quotients therefore realizes every finite p-group as the Galois group of a totally ramified extension with residue field ${\mathbb{F}}_p$.

For an integral domain $A$ with fraction field $K$, realize Cartier divisors as invertible fractional ideals, with local divisor equation $f$ corresponding to the ideal $fA$. This is the $\mathcal{O}(-D)$ convention. The maps in the displayed exact sequence are unit inclusion, $f\mapsto fA$, and $I\mapsto [I]$. A principal fractional ideal is trivial as a divisor exactly when its generator is a unit. An invertible fractional ideal is trivial in the Picard group of a ring exactly when it is principal. Every rank-one projective module embeds in $K$ and produces an invertible fractional ideal, proving surjectivity. Local multiplication identifies $I{\otimes }_{A}J$ with $IJ$, so these are group homomorphisms.

Passing to norm limits in Artin reciprocity gives an exact sequence connecting closures of global units, local pro-$p$ units at primes over $p$, the p-ramified Iwasawa module and the unramified Iwasawa module. It explains arithmetic Iwasawa-module ranks by comparing local and global unit ranks. Roots of unity and the precise splitting conventions must be retained when seeking exact integral statements.

Let $V_{\infty }$ be the norm limit of local principal units in the $p$-power cyclotomic field tower, for odd $p$. The Coleman power series $f_u$ gives the Frobenius-corrected logarithm $p^{-1}\log (f_u^p/\varphi f_u)$, whose inverse Amice transform is supported on ${\mathbb{Z}}_p^{\times }$. This is the middle map in the displayed exact sequence; the last map is the first moment $\mu \mapsto \int xd\mu$. Its kernel consists of norm-compatible $p$-power roots of unity. The logarithmic difference equation has one resonant degree, namely degree one in $\log (1+T)$, accounting for the last term. Complex conjugation acts by $-1$ on both copies of ${\mathbb{Z}}_p(1)$. Taking its even part therefore identifies the local principal-unit limit in the real tower with ${\mathbb{Z}}_p[[{\mathbb{Z}}_p^{\times }/\{\pm 1\}]]$.

An exact sequence of sheaves of abelian groups is a sequence of sheaf homomorphisms for which the image of each map equals the kernel of the next. Exactness can be checked on every stalk of a sheaf. Exactness of the associated sequences of sections on every open set is a stronger condition: a surjective sheaf homomorphism need only admit lifts locally.

Over $\mathbb{C}$, the four partial derivatives show that this projective surface is smooth. Adjunction makes its canonical bundle trivial, and the hypersurface exact sequence gives $H^1({\mathcal{O}}_X)=0$, so it is a K3 surface. Lines given by $x_0=\zeta x_1$, $x_2=\eta x_3$, with ${\zeta }^4={\eta }^4=-1$, lie on it. Choosing both roots differently gives skew lines, one of which is a section of the elliptic pencil defined by planes through the other.

A module is finitely presented when it admits the displayed exact sequence for finite integers $r,s$. Thus it has finitely many generators and finitely many relations. On a Noetherian ring, every finitely generated module has a finite presentation.

For commuting five-term exact sequences in an abelian category, the first vertical map being epic and the second invertible give an invertible induced map on the images entering the middle objects, because these images are the cokernels of the first horizontal maps. The fourth vertical map being invertible and the fifth monic similarly give an invertible map on the images leaving the middle objects, because these are the kernels of the last horizontal maps. The image factorization in an abelian category packages each middle object between these two images in a short exact sequence. The short five lemma then makes the middle vertical map invertible. The proof is categorical and does not require an embedding into a category of modules.

A flasque resolution is an exact sequence of sheaves of abelian groups whose terms ${\mathcal{I}}^j$ are flasque sheaves. Every such sheaf embeds by its section germs into the sheaf $U\mapsto {\prod }_{P\in U}{\mathcal{F}}_P$. Its restriction maps are projections, so it is flasque. Apply this construction successively to the cokernels to obtain a resolution. The sheaf cohomology of $\mathcal{F}$ is the cohomology of the cochain complex $\Gamma (X,{\mathcal{I}}^{\bullet })$. A quotient of two flasque sheaves in a short exact sequence of sheaves is flasque: the flasque kernel allows local lifts to be glued into a section of the middle sheaf over the smaller open set; extend that lift using flasqueness of the middle sheaf, and project. Consequently the resolution's successive cokernels are flasque when $\mathcal{F}$ is flasque, and exactness of sections shows that all its positive-degree cohomology vanishes.

For a vector bundle of finite positive rank let $B$ be an upper degree bound for the line subbundles of $E^{\ast }\otimes K_C$. If $m-1>B$, no nonzero map ${\mathcal{O}}_C(m{p}_0-p)\to E^{\ast }\otimes K_C$ can exist for any point $p$: its saturated image would have degree at least $m-1$. Serre duality then gives $H^1(E(m{p}_0-p))=0$. The evaluation exact sequence shows that $H^0(E(m{p}_0))\to E(m{p}_0){|}_p$ is surjective at every point, proving the assertion with a uniform bound in $p$.

For a finite free module $M$ of rank $n$, this scheme represents rank-$r$ locally free module quotients of $M$ after base change, with quotient maps identified up to commuting isomorphism. It carries a universal exact sequence $0\to \mathcal{S}\to M\otimes \mathcal{O}\to \mathcal{Q}\to 0$, of ranks $n-r,n,r$. Its standard affine charts of the quotient Grassmannian prove representability over any base ring. Over a field, taking kernels gives the usual Grassmannian of $(n-r)$-planes.

// Target: ringed-space.bigb

An injective resolution is an exact sequence starting with a sheaf and continuing with injective sheaves. It is constructed by successively embedding each quotient in an injective sheaf. Taking global sections and then cohomology computes sheaf cohomology; in particular an injective sheaf has zero higher cohomology.

Let $L$ be the complexification of a real vector bundle of the tautological bundle over Real projective space ${\mathbb{RP}}^{2n+1}$. For $k\ge 1$, the K-theory Euler class of $L^{\oplus k}$ is $-2^{k-1}([L]-1)$, andThe K-theory Gysin sequence of a sphere bundle gives an extension by $\mathbb{Z}$, which splits as an exact sequence of abelian groups.

For a pair $(X,A)$ of CW complexes, relative topological K-theory and Bott periodicity give the cyclic exact sequence

A locally free resolution of a sheaf of modules $F$ is an exact sequence $\cdots \to G_2\to G_1\to G_0\to F\to 0$ with each $G_i$ a locally free sheaf. If every coherent sheaf is the quotient of a locally free sheaf of finite rank on a noetherian scheme, iteratively taking a kernel sheaf constructs such a resolution with coherent sheaves at every stage. The kernel sheaves need not be locally free sheaves, so a finite resolution requires additional hypotheses. A uniform bound on sheaf cohomology can nevertheless make a finite initial segment sufficient for dimension shifting in sheaf cohomology.

All the special orthogonal groups are connected, and their second homotopy groups vanish. The rotation circle gives $SO(2)\cong S^1$. Conjugation by unit quaternions gives the twofold universal cover $S^3\to SO(3)$, so ${\pi }_{1}SO(3)=\mathbb{Z}/2$ and ${\pi }_{3}SO(3)=\mathbb{Z}$. The special orthogonal sphere fibration $SO(n-1)\to SO(n)\to S^{n-1}$ propagates the first two homotopy groups for $n\ge 4$. In dimension four its exact sequence has a zero map from the torsion group ${\pi }_4{S}^3=\mathbb{Z}/2$ to ${\pi }_{3}SO(3)=\mathbb{Z}$, giving ${\pi }_{3}SO(4)={\mathbb{Z}}^2$. For $SO(1)$ all positive groups vanish, and ${\pi }_{3}SO(2)=0$.

Take the compact Riemann surface $X={\mathbb{P}}^1$ and the holomorphic vector bundle $E={\mathcal{O}}_X\oplus {\mathcal{O}}_X$. For every positive integer $m$, use homogeneous coordinates $[Z_0:Z_1]$ and the mapThe two sections $Z_0^m,Z_1^m$ never vanish together on ${\mathbb{P}}^1$. Thus this map has rank one at every point; locally one of its two components is invertible in suitable line-bundle frames, so its image is a genuine holomorphic subbundle and its quotient is a locally free sheaf. More explicitly, there is an exact sequenceExactness follows fibrewise from the absence of a common zero and hence also locally as a sequence of vector bundles. The image $F_m$ is isomorphic to ${\mathcal{O}}_X(-m)$ and has rank one, soThus there need not be a lower bound for subbundle slopes, even for a fixed trivial rank-two bundle. Merely multiplying a section of one fixed subline by a function with zeros would not suffice: that could give only a subsheaf with a torsion quotient. The nowhere-simultaneously-zero pair above avoids that problem and realizes unbounded negative degrees of line subbundles.

At $2$ there is good reduction, and the surjectivity of good reduction over a local field gives an exact sequenceThe quotient has order $5$ and is consequently a cyclic group. For every odd integer $n$, multiplication by $n$ is an automorphism of $E_1({\mathbb{Q}}_2)$: its formal group law has multiplication series $nt+O(t^2)$ with unit linear coefficient, whose inverse formal power series has integral coefficients and converges on $2{\mathbb{Z}}_2$. This is the prime-to-residue-characteristic multiplication on a formal group.

Thus the odd-primary torsion subgroup injects into the order-five quotient. It also realizes that quotient: choose a lift $P$ of any reduced point, so $[5]P\in E_1$. There is a unique $Q\in E_1$ with $[5]Q=[5]P$. Then $P-Q$ is killed by five and has the prescribed reduction. This proves the prime-to-p torsion lifts uniquely at good reduction directly here. Therefore, for an odd prime $p$,The five-primary subgroup consists of the unique five-torsion lifts of the five reduced points. These lifts need not be rational over $\mathbb{Q}$; the preceding answer about rational torsion points of an elliptic curve does not contradict their existence over the p-adic field.

A precise version of the Hilbert-Serre theorem is as follows. Let $R={⨁}_{j\ge 0}{R}_j$ be a commutative graded ring, generated over an Artinian degree-zero ring $R_0$ by homogeneous elements $y_1,\dots ,y_s$ of positive degrees $d_1,\dots ,d_s$. Let $M$ be a finitely generated graded module, bounded below. Each component has finite module length over $R_0$. Its Poincare series of a graded module, or Hilbert series, isWhen $R_0$ is a field, these lengths are dimensions. The theorem assertsFor a nonnegatively graded module the numerator can be taken in $\mathbb{Z}[t]$. The finite-generation and finite-length hypotheses specify the meaning of the series; they are part of the theorem's statement.

We prove it by induction on $s$. Regard $M$ as a finite graded module over $S=R_0[Y_1,\dots ,Y_s]$, with $\mathrm{deg}{Y}_i=d_i$, acting through the specified generators of $R$. This polynomial ring is Noetherian by the Hilbert basis theorem and Question 1. When $s=0$, finitely many homogeneous generators over $R_0$ mean that only finitely many components are nonzero, so the series is a Laurent polynomial. For the induction step, put $y=Y_s$, $d=d_s$, $K=(0{:}_{M}y)$ and $C=M/yM$. Both $K$ and $C$ are finite graded modules; $y$ annihilates them, so they are modules over $R_0[Y_1,\dots ,Y_{s-1}]$. The finite generation of $K$ uses Noetherianity of $S$.

The exact sequence of graded modulesgives, by additivity of component lengths,Here the shift is defined so that $P_{M(-d)}=t^d{P}_M$. By induction both series on the right have denominator ${\prod }_{i<s}(1-t^{d_i})$. Dividing by $1-t^d$ proves the result. The annihilator term cannot be omitted unless multiplication by $y$ is injective.

In the standard grading $d_i=1$, the coefficients of $(1-t{)}^{-s}$ are $\left(\genfrac{}{}{0ex}{}{j+s-1}{s-1}\right)$. Multiplying by the finite numerator shows that the component dimensions agree with a polynomial for sufficiently large $j$. Cumulative dimensions have generating function $P_M(t)/(1-t)$ and are also eventually polynomial. This consequence supplies the dimension used for Weyl algebra modules.

Use the Bernstein filtration $B_m{A}_n$, in which both coordinate generators and differentiation generators have degree one. The ordered monomial basis of a Weyl algebra givesChoose a finite-dimensional generating subspace $M_0$ of a nonzero finitely generated left module $M$ and set $M_m=B_m{A}_n{M}_0$. This is a good filtration of a module: the symbols of the generators generate its associated graded module over the displayed polynomial ring. The Hilbert-Serre theorem makes ${\dim }_{\mathbb{C}}{M}_m$ an eventual polynomial. Define the Bernstein growth dimension of a Weyl algebra module byThe dimension is independent of the chosen finite generating subspace. Each of two finite generating sets is contained in a fixed filtration step for the other, giving $M_m\subseteq M_{m+c}^{\prime }$ and $M_m^{\prime }\subseteq M_{m+c^{\prime }}$. Such fixed shifts preserve the degree of growth. The same argument applies to any good Bernstein filtration. This dimension agrees with the Gelfand–Kirillov dimension of a module; one may set $d(0)=-\infty$.

Let the nonconstant element $D$ have Bernstein degree $d\ge 1$. Since ${\mathrm{gr}}_B{A}_n$ is a domain, the product of two nonzero symbols cannot vanish. HenceRight multiplication by $D$ is injective, using the domain property of $A_n$. Thus the quotient filtration on $M=A_n/A_{n}D$ has cumulative dimensionsThe leading terms of degree $2n$ cancel, and the leading surviving term isIt is nonzero, because $d\ge 1$. ThereforeEquivalently the Hilbert series of the associated graded quotient is $(1-t^d)/(1-t{)}^{2n}$. Here $d$ is Bernstein degree, not differential order: multiplication by $x_1$ has differential order zero but Bernstein degree one. Using the order filtration in this dimension argument would lose the finite-dimensional filtration pieces.

Let $C$ be the smooth algebraic curve. A nonzero regular section of a line bundle has an effective zero divisor, so its degree of a line bundle is nonnegative. Degrees add under tensor products and change sign on taking the dual. We use genuine line subbundles, so their quotients in $E$ are locally free. Both bounds can be proved constructively.

For the upper bound, take a rational frame of $E^{\ast }$ over the function field of $C$. Its two members are rational sections with only finitely many poles. Choose an effective divisor $D$ large enough to bound the pole orders of both rational sections. They become global sections of $E^{\ast }\otimes {\mathcal{O}}_C(D)$ and define a sheaf mapwhich is an isomorphism at the generic point. It is therefore injective as a sheaf map: a section in its kernel would vanish generically in a locally free sheaf, which is a torsion-free sheaf. For any line subbundle $L\subset E$, at least one of its two component maps into ${\mathcal{O}}_C(D)$ is nonzero. Such a map is a nonzero global section of ${\mathcal{O}}_C(D)\otimes L^{-1}$. Its zero divisor on an algebraic curve is effective, soThis upper degree bound for line subbundles on a smooth curve depends only on a choice made for $E$, and applies to every one of its line subbundles.

For the lower-bound question we will actually construct subbundles ${\mathcal{O}}_C(-m{p}_0)$ for every sufficiently large integer $m$, where $p_0$ is a fixed point. First prove the globally generated vector bundle property rather than assuming that negative-degree inclusions automatically are subbundles.

Apply the upper-bound argument to the rank-two vector bundle $F=E^{\ast }\otimes K_C$, where $K_C$ is the canonical bundle. Let $B$ be its upper degree bound. Choose $m$ with $m-1>B$. For any point $p\in C$, a nonzero global section of $F(-m{p}_0+p)$ would give a nonzero mapIts image can be saturated to a saturated line subbundle on a smooth curve $M\subset F$. Locally the local ring of a smooth algebraic curve is a discrete valuation ring: dividing out the common power of a uniformizer makes the image a primitive rank-one summand, and its quotient is free. Thus the saturation is indeed a subbundle. The original map gives a nonzero global section of $M\otimes {\mathcal{O}}_C(-m{p}_0+p)$ and consequently $\mathrm{deg}M\ge m-1$, contradicting the upper bound. Hence $H^0(F(-m{p}_0+p))=0$ for every $p$. By Serre duality,The exact sequenceand its long exact sequence in sheaf cohomology show that evaluation $H^0(E(m{p}_0))\to E(m{p}_0){|}_p$ is onto for every $p$. This proves the global generation of positive twists on a smooth curve with one bound on $m$ that works at every point.

Write $W=H^0(E(m{p}_0))$ and $N=\dim W$. The evaluation map $W\otimes {\mathcal{O}}_C\to E(m{p}_0)$ is surjective with kernel bundle of rank $N-2$. Consider the incidence subsetIt is the total space of this kernel bundle, so $\dim Z=1+(N-2)=N-1$. Its projection into $W$ is closed because $C$ is proper, and its image has dimension at most $N-1<N$. Choose $s$ outside this proper closed subset. It vanishes nowhere. In a local trivialization at any point, one component of $s$ is a unit, so it can be completed to a basis. Therefore ${\mathcal{O}}_C\stackrel{s}{\to }E(m{p}_0)$ is a subbundle with a locally free rank-one quotient. Twisting back givesThis proves the unbounded negative degrees of line subbundles. The nowhere-vanishing construction guarantees a subbundle at every point, including the zeros that might otherwise have occurred in a generic meromorphic inclusion. If the curve has several connected components, apply these constructions on each component and add the degrees; the same upper-bound and no-lower-bound conclusions follow.